| The Abstractor | Elementary Notions | A Fourth Dimension | A Division Algebra | Closure | An Example of Hamilton's Work | Bibliography | Back to the front page A Division Algebra Joseph Johnson
Hamilton had discovered what modern mathematicians term a skew field. That is, the system satisfies the following
axioms applied to the direct sum of four real additive groups, under Hamilton's definitions of equality, addition, and
multiplication. Let be any quaternions with
where . This can be derived as a
consequence of conjugacy in Quaternions (see Elements Paragraph 187). Every quaternion has a conjugate
such that is a real number. This allows us to set .
The explicit conjugate of is (which is
analogous to conjugacy of complex numbers).
If this axiom did not hold, the Quaternions would be utterly uninteresting. It would represent two separate group structures
(after deleting 0 in the multiplicative case) - hardly impressive to most mathematicians.
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